Abstract

The asymptotic methods developed by Horwitz for the analysis of strip unstable laser resonators are extended to resonators with circular mirrors. The theory is developed and shown to give results in good agreement with results obtained using more conventional techniques, with substantial savings in computer time. The interleaving of the resonator eigenvalue graphs is seen to persist, even at very large Fresnel numbers. Further applications of asymptotic resonator analysis are suggested.

© 1978 Optical Society of America

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References

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  1. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
    [Crossref]
  2. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
    [Crossref]
  3. P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [Crossref]
  4. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
    [Crossref] [PubMed]
  5. A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2736 (1970).
    [Crossref] [PubMed]
  6. F. W. J. Olver, “Error bounds for stationary phase approximations,” SIAM J. Math. Anal. 5, 19–29 (1974).
    [Crossref]
  7. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. GPO, Washington, D.C., 1964).
  8. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [Crossref] [PubMed]
  9. P. Horwitz, “Modes in misaligned unstable resonators,” Appl. Opt. 15, 167–178 (1976).
    [Crossref] [PubMed]
  10. G. T. Moore and R. J. McCarthy, “Theory of modes in a loaded strip confocal unstable resonator,” J. Opt. Soc. Am. 67, 228–241 (1977).
    [Crossref]

1977 (1)

1976 (1)

1974 (2)

F. W. J. Olver, “Error bounds for stationary phase approximations,” SIAM J. Math. Anal. 5, 19–29 (1974).
[Crossref]

A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13, 353–367 (1974).
[Crossref] [PubMed]

1973 (1)

1970 (1)

1965 (1)

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[Crossref]

1962 (1)

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[Crossref]

Horwitz, P.

Keller, J. B.

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[Crossref]

McCarthy, R. J.

Miller, H. Y.

Moore, G. T.

Olver, F. W. J.

F. W. J. Olver, “Error bounds for stationary phase approximations,” SIAM J. Math. Anal. 5, 19–29 (1974).
[Crossref]

Siegman, A. E.

Appl. Opt. (3)

Bell Sys. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[Crossref]

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[Crossref]

SIAM J. Math. Anal. (1)

F. W. J. Olver, “Error bounds for stationary phase approximations,” SIAM J. Math. Anal. 5, 19–29 (1974).
[Crossref]

Other (1)

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. GPO, Washington, D.C., 1964).

Cited By

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Figures (9)

FIG. 1
FIG. 1

Equivalent lens train for a symmetric unstable resonator.

FIG. 2
FIG. 2

Magnitude of λ vs Neq for M = 2, I = 0, 1 ≤ Neq ≤ 10. The solid curves were produced using the asymptotic analysis. The dashed curves are from Ref. 5.

FIG. 3
FIG. 3

Magnitude of λ vs Neq for M = 2, I = 0, 10 ≤ Neq ≤ 20.

FIG. 4
FIG. 4

Magnitude of λ vs Neq for M = 2, I = 0, 20 ≤ Neq ≤ 30.

FIG. 5
FIG. 5

Magnitude of λ vs Neq for M = 2, I = 1, 20 ≤ Neq ≤ 30.

FIG. 6
FIG. 6

Magnitude of λ vs Neq for M = 2, I = 2, 30 ≤ Neq ≤ 40.

FIG. 7
FIG. 7

Magnitude of λ vs Neq for M = 2, I = 0, 1000 ≤ Neq ≤ 1010. The stationary roots have been suppressed.

FIG. 8
FIG. 8

|F(x,t)|2 vs x for t = 20 and 0 ≤ x ≤ 1. Solid curve was obtained by using Eq. (13) to define F; dotted curve obtained by using Eq. (29).

FIG. 9
FIG. 9

Irradiance profile of dominant mode for M = 2, I = 0, and Neq = 6.5. Mirror edge is at 1, shadow boundary is at 2. Solid curve was obtained from a conventional iterative calculation; dotted curve was obtained using asymptotic analysis.

Equations (53)

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γ ψ ( x ) = i l + 1 2 π N 0 0 1 y J l ( 2 π N 0 x y ) × exp ( - i π N 0 2 M ( M 2 + 1 ) ( x 2 + y 2 ) ) ψ ( y ) d y .
γ 1 / M ,
ψ ( x ) exp ( - i π N eq x 2 ) ,
ψ ( x ) = f ( x ) exp ( - i π N eq x 2 ) ,
λ f ( x ) = i l + 1 2 t 0 1 y J l ( 2 t x y / M ) × exp { - i t [ y 2 + ( x / M ) 2 ] } f ( y ) d y ,
I = a b q ( y ) e - i t p ( y ) d y ,
I ~ e - i π / 4 q ( y 0 ) e - i t p ( y 0 ) 2 π / t p ( y 0 ) 1 / 2 + i t [ q ( b ) p ( b ) e - i t p ( b ) - q ( a ) p ( a ) e - i t p ( a ) ] .
p ( y ) p ( b ) + p ( b ) ( y - b ) .
p ( y ) p ( b ) + p ( b ) ( y - b ) + ½ p ( b ) ( y - b ) 2
J l ( z ) ~ ( 2 / π z ) 1 / 2 cos [ z - l π / 2 - π / 4 ]
f ( x / M ) - i l f ( 1 ) 1 - ( x / M ) 2 exp { - i t [ 1 + ( x M ) 2 ] } × ( M / π x t ) 1 / 2 [ cos ( 2 t x / M - l π / 2 - π / 4 ) + i ( x / M ) sin ( 2 t x / M - l π / 2 - π / 4 ) ] .
F 1 ( x ) = - i l exp { - i t [ 1 + ( x / M ) ] 2 } 1 - ( x / M ) 2 × [ J l ( 2 t x / M ) + i ( x / M ) J l + 1 ( 2 t x / M ) ] .
U ( r ) = - ( i K / z ) e - i K r 2 / 2 z 0 R e i K ρ 2 / 2 z J 0 ( K r ρ z ) ρ d ρ ,
U ( r ) = 1 - e i π N 0 + e i π N 0 ( π N 0 ) 2 ( r / R ) 2 + ,
U ( r ) ~ 1 - e i π N 0 + e i π N 0 [ ( π N 0 ) 2 - 1 ] ( r / R ) 2 + .
F ( x , t ) = - [ i l e - i t ( 1 + x 2 ) / ( 1 - x 2 ) ] × [ J l ( 2 t x ) + i x J l + 1 ( 2 t x ) ] .
M n = k = 0 n M - 2 k .
F n ( x ) = F ( x / M n , t / M n - 1 ) ,             n = 1 , 2 , .
I n = i l + 1 2 t 0 1 y J l ( 2 t x y M ) e - i t [ y 2 + ( x / M ) 2 ] F n ( y ) d y .
I n = ( SP ) n + ( EP ) n ,
( SP ) n = F n + 1 ( x ) ;
( EP ) n F n ( 1 ) F 1 ( x ) .
f ( x ) = 1 + n = 1 N a n F n ( x ) ,
λ ( 1 + n = 1 N a n F n ( x ) ) = 1 + n = 1 N a n F n + 1 ( x ) + ( 1 + n = 1 N a n F n ( 1 ) ) F 1 ( x ) .
a n + 1 = a n / λ = a 1 / λ n ,
λ - 1 = a N F N + 1
a n = [ ( λ - 1 ) / F N + 1 ] λ N - n .
λ a 1 = 1 + n = 1 N a n F n ( 1 ) .
λ N ( λ - 1 ) = F N + 1 + ( λ - 1 ) n = 1 N λ N - n F n ( 1 ) .
f ( x ) = n = 1 N a n F n ( x ) .
λ N = n = 1 N λ N - n F n ( 1 )
a n = λ - n .
F ( x , t ) = [ i l + 1 / ( 2 x ) 1 / 2 ] { e - i ( l π / 2 + π / 4 ) × [ E * ( [ 2 t / π ] 1 / 2 [ 1 - x ] ) - ( 1 - i ) / 2 ] + e i ( l π / 2 + π / 4 ) [ E * ( ( 2 t / π ) 1 / 2 ( 1 + x ) ) - ( 1 - i ) / 2 ] } ,
f ( x ) = a 1 F 1 ( x ) ,
F 1 ( x ) = i l + 1 ( M / 2 x ) 1 / 2 { e - i ( l π / 2 + π / 4 ) × [ E * ( [ 2 t / π ] 1 / 2 [ 1 - x / M ] ) + ( 1 - i ) / 2 ] + e i ( l π / 2 + π / 4 ) [ E * ( [ 2 t / π ] 1 / 2 [ 1 + x / M ] ) - ( 1 - i ) / 2 ] } .
F ( x , t ) = - i l exp { - i t [ 1 + x 2 ] } ( 1 - x 2 ) - 1 × [ J l ( 2 t x ) + i x J l + 1 ( 2 t x ) ] , F n ( x ) = F ( x / M n , t / M n - 1 ) ,             n = 1 , 2 , .
I n = 2 t i l + 1 0 1 y exp { - i t [ y 2 + ( x / M ) 2 ] } × J l ( 2 t x y / M ) F n ( y ) d y ; I n = 2 i t ( - 1 ) l + 1 0 1 y J l ( 2 t x y / M ) exp { - i t [ y 2 + ( x / M ) 2 ] } × exp { - ( i t / M n - 1 ) [ 1 + ( y / M n ) 2 ] } [ 1 - ( y / M n ) 2 ] - 1 × { J l ( 2 t y / M n - 1 M n ) + ( i y / M n ) J l + 1 ( 2 t y / M n - 1 M n ) } d y .
I n ~ ( - 1 ) l + 1 ( i / 2 π ) ( M n + 1 M n - 1 / x ) 1 / 2 × exp { - i t [ 1 / M n - 1 + ( x / M ) 2 ] } × { - i ( - 1 ) l K 1 + K 2 + K 3 + i ( - 1 ) l K 4 } ,
K s = 0 1 q s ( y ) e - i t P s ( y ) d y ;
q 1 ( y ) = q 2 ( y ) = ( 1 - y / M n ) - 1 ;
q 3 ( y ) = q 4 ( y ) = ( 1 + y / M n ) - 1 ;
P 1 ( y ) = ( M n / M n - 1 ) y 2 - 2 y ( x / M + 1 / M n M n - 1 ) ;
P 2 ( y ) = ( M n / M n - 1 ) y 2 + 2 y ( x / M - 1 / M n M n - 1 ) ;
P 3 ( y ) = ( M n / M n - 1 ) y 2 + 2 y ( 1 / M n M n - 1 - x / M )
P 4 ( y ) = ( M n / M n - 1 ) y 2 + 2 y ( x / M + 1 / M n M n - 1 ) .
exp { - i t [ 1 / M n - 1 + ( x / M ) 2 ] } SP ( K 1 ) = e - i π / 4 ( π M n / t M n - 1 ) 1 / 2 ( 1 - x / M n + 1 ) - 1 × exp { - ( i t / M n ) [ 1 - x / M n + 1 ) 2 } .
( SP ) n ~ - i l ( M n + 1 M n / π t x ) 1 / 2 × exp { - ( i t / M n ) [ 1 + ( x / M n + 1 ) 2 ] } [ 1 - ( x / M n + 1 ) 2 ] - 1 × [ cos ( 2 t x / M n + 1 M n - l π / 2 - π / 4 ) + i ( x / M n + 1 ) × sin ( 2 t x / M n + 1 M n - l π / 2 - π / 4 ) ] .
EP ( K 1 ) = ( i / 2 t ) ( 1 - M - n ) - 1 ( d n - - x / M ) - 1 × exp { - i t [ M n / M n - 1 - 2 ( x / M + 1 / M n M n - 1 ) ] } ;
EP ( K 2 ) = ( i / 2 t ) ( 1 - M - n ) - 1 ( d n - + x / M ) - 1 × exp { - i t [ M n / M n - 1 + 2 ( x / M - 1 / M n M n - 1 ) ] } ;
EP ( K 3 ) = ( i / 2 t ) ( 1 + M - n ) - 1 ( d n + - x / M ) - 1 × exp { - i t [ M n / M n - 1 + 2 ( 1 / M n M n - 1 - x / M ) ] } ;
EP ( K 4 ) = ( i / 2 t ) ( 1 + M - n ) - 1 ( d n + + x / M ) - 1 × exp { - i t [ M n / M n - 1 + 2 ( x / M + 1 / M n M n - 1 ) ] } ;
d n ± = ( M n ± M - n ) / M n - 1 .
( EP ) n ~ ( - 1 ) l 4 π t ( M n + 1 M n - 1 x ) 1 / 2 exp { - i t [ M n + 1 M n - 1 + ( x M ) 2 ] } { - i ( - 1 ) l exp { - i t [ - 2 x / M - 2 / M n M n - 1 ] } ( 1 - 1 / M n ) ( 1 - x / M ) + exp { - i t [ 2 x / M - 2 / M n M n - 1 ] } ( 1 - 1 / M n ) ( 1 + x / M ) + exp { - i t [ - 2 x / M + 2 / M n M n - 1 ] } ( 1 + 1 / M n ) ( 1 - x / M ) + i ( - 1 ) l exp { - i t [ 2 x / M + 2 / M n M n - 1 ] } ( 1 + 1 / M n ) ( 1 + x / M ) } = ( - 1 ) l 4 π t ( M n + 1 M n - 1 x ) 1 / 2 exp { - i t [ 1 + M n M n - 1 + ( x M ) 2 ] } { e i ( 2 t x / M - l π / 2 - π / 4 ) 1 - x / M + e - i ( 2 t x / M - l π / 2 - π / 4 ) 1 + x / M } × { e i ( 2 t / M n M n - 1 - l π / 2 - x / 4 ) 1 - 1 / M n + e - i ( 2 t / M n M n - 1 - l π / 2 - π / 4 1 + 1 / M n } = ( - 1 ) l π t ( M n + M n - 1 x ) 1 / 2 e - i t [ 1 + ( x / M ) 2 ] 1 - ( x / M ) 2 e - i t [ 1 + M - 2 n ] 1 - M - 2 n × [ cos ( 2 t x / M - l π / 2 - π / 4 ) + i ( x / M ) sin ( 2 t x / M - l π / 2 - π / 4 ) ] × [ cos ( 2 t / M n M n - 1 - l π / 2 - π / 4 ) + ( i / M n ) sin ( 2 t / M n M n - 1 - l π / 2 - π / 4 ) ] .